Mathematical Physics, Analysis and Geometry - Volume 9 by V. A. Marchenko, A. Boutet de Monvel, H. McKean (Editors)

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By V. A. Marchenko, A. Boutet de Monvel, H. McKean (Editors)

Articles during this volume:

Semiclassical Weyl formulation for a category of Weakly standard Elliptic Operators
Lech Zielinski

The PDEs of Biorthogonal Polynomials bobbing up within the Two-Matrix Model
Marco Bertola and Bertrand Eynard

Sharpenings of Li's Criterion for the Riemann Hypothesis
André Voros

Persistence of Eigenvalues and Multiplicity within the Dirichlet challenge for the Laplace Operator on Nonsmooth Domains
Pier Domenico Lamberti and Massimo Lanza de Cristoforis

Geometry and progress price of Frobenius Numbers of Additive Semigroups
V. I. Arnold

Singular Spectrum close to a novel element of Friedrichs version Operators of Absolute Type
Serguei I. Iakovlev

Deformations of Fuchsian structures of Linear Differential Equations and the Schlesinger System
Victor Katsnelson and Dan Volok

Gaussian Beam development for Adiabatic Perturbations
M. Dimassi, J.-C. Guillot and J. Ralston

A Geometrical Interpretation of ‘Supergauge’ changes utilizing D -Differentiation
D. J. Hurley and M. A. Vandyck

Reflection in a Translation Invariant Surface
Brendan Guilfoyle and Wilhelm Klingenberg

On Separation of Variables for Homogeneous SL(r) Gaudin Systems
Gregorio Falqui and Fabio Musso

On the 2 Spectra Inverse challenge for Semi-infinite Jacobi Matrices
Luis O. Silva and Ricardo Weder

The cover Graph and point statistics for Random Operators on Trees
Michael Aizenman and Simone Warzel

Waveguides with mixed Dirichlet and Robin Boundary Conditions
P. Freitas and D. Krejčiřík

Generalized Weierstrass relatives and Frobenius Reciprocity
Shigeki Matsutani

Form-preserving alterations for the Time-dependent Schrödinger Equation in ( n + 1) Dimensions
Axel Schulze-Halberg

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Extra resources for Mathematical Physics, Analysis and Geometry - Volume 9

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N $ ( À i=2 k farg ( >0g k ½RH falseŠ; ð26Þ which gives oscillations that blow up exponentially with n. , min farg (k >0g fjIm 1=(k jÀ1p ffi Sn can only reliably signal a zero violating RH up to a height jIm &j | n=2; so vice-versa, since such zeros are now known to require jIm &j a 109 , they could only be detected by Sn for n a 1018 (see also [1, 23], [14, p. 5]). Our asymptotic criteria may thus not surpass others in practical sensitivity, but their sharpness is theoretically interesting. n ! 0 are ruled out.

1 À Ád2 0 1 J À1 Tr P 1 À Ae  à ; V 2 ð PÞ À x Å vd2 þ1 NÀ1 ¼ h i 1 À Á Tr P J ð1 À RJ ÞÀ1  À1 Ã; V20 ðPÞ À x Å NÀ1 V200 yj d2 X j¼1 ¼ d2 X 1 À Á Tr P J ð1 À RJ ÞÀ1  À1 ÃðV20 ðPÞ À xÞ Å À NÀ1 V200 yj j¼1 À ¼ d2 X 00 j¼1 V 2 d2 X j¼1 À ¼1 zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ À Á V 0 ðPÞ À x 1 À Á Tr P J ð1 À RJ ÞÀ1  À1 à P À yj 2 Å þ P À yj NÀ1 V200 yj d2 X j¼1 ¼ d2 X j¼1 À ¼   À ÁÀ ÁÀ1 À1 1 V 0 ð PÞ À x t À1 À Á Tr P J 2 à  à þ  ÃÅ 1 À R P À y Å j j P À yj 0 NÀ1 V200 yj d2 X 1 1 À Á Tr P J Sj ðPÞ Å À À Á Tr PJ SJ ðPÞÃt à ŠÀ 00 00 NÀ1 NÀ1 V2 yj V j¼1 2 yj d2 X 00 j¼1 V 2 d2 X j¼1 1 À Á Tr P J ðV20 ðPÞ À xÞð1 À RJ ÞÀ1  À1 à ŠNÀ1 yj À ÁÀ Á 1 À Á Tr P J SJ ðPÞÅ P À yj 1 À Rj À1  À1 à Š0 NÀ1 yj d2 X 1 1 À Á Tr P J SJ ðPÞÅ À À Á Tr P J Sj ðPÞÅ Â 00 y 0 0 V200 yj V j¼1 2 j À ÁÀ ÁÀ1  P À yj 1 À Rj  À1 à ŠNÀ1 ð4:27Þ 46 MARCO BERTOLA AND BERTRAND EYNARD Notice that if N > 2d2 the PNj 1 in the last trace is irrelevant and therefore that trace is independent of N.

All of that is realized in many cases including, but not limited to, Dedekind zeta functions [13, 15, 27] and some Dirichlet L-functions [5, 18, 27]; see also [14] (discussed at end); in all those instances, NðTÞ ¼ Oðlog TÞ. n , our saddle-point evaluation then always yields: either (16) ) (15) if all the zeros have Re & ¼ 12 Y or the immediately general result (14) otherwise. Like (14) before, (15) can be derived quite rigorously but by still another method, previously unknown to us, and written for the Riemann zeros by J.

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